• 제목/요약/키워드: Polar Derivative

검색결과 38건 처리시간 0.022초

GENERALIZATION OF SOME INEQUALITIES TO THE CLASS OF GENERALIZED DERIVATIVE

  • Wani, Irfan Ahmad;Mir, Mohammad Ibrahim;Nazir, Ishfaq
    • 한국수학교육학회지시리즈B:순수및응용수학
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    • 제28권4호
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    • pp.343-353
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    • 2021
  • In this paper, we obtain some inequalities concerning the class of generalized derivative and generalized polar derivative which are analogous respectively to the ordinary derivative and polar derivative of polynomials.

SOME INEQUALITIES ON POLAR DERIVATIVE OF A POLYNOMIAL

  • N., Reingachan;Robinson, Soraisam;Barchand, Chanam
    • Nonlinear Functional Analysis and Applications
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    • 제27권4호
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    • pp.797-805
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    • 2022
  • Let P(z) be a polynomial of degree n. A well-known inequality due to S. Bernstein states that if P ∈ Pn, then $$\max_{{\mid}z{\mid}=1}\,{\mid}P^{\prime}(z){\mid}\,{\leq}n\,\max_{{\mid}z{\mid}=1}\,{\mid}P(z){\mid}$$. In this paper, we establish some extensions and refinements of the above inequality to polar derivative and some other well-known inequalities concerning the polynomials and their ordinary derivatives.

SOME INEQUALITIES ON POLAR DERIVATIVE OF A POLYNOMIAL

  • Devi, Khangembam Babina;Krishnadas, Kshetrimayum;Chanam, Barchand
    • Nonlinear Functional Analysis and Applications
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    • 제27권1호
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    • pp.141-148
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    • 2022
  • Let p(z) be a polynomial of degree n having no zero in |z| < k, k ≤ 1, then Govil proved $$\max_{{\mid}z{\mid}=1}{\mid}p^{\prime}(z){\mid}{\leq}{\frac{n}{1+k^n}}\max_{{\mid}z{\mid}=1}{\mid}p(z){\mid}$$, provided |p'(z)| and |q'(z)| attain their maximal at the same point on the circle |z| = 1, where $$q(z)=z^n{\overline{p(\frac{1}{\overline{z}})}}$$. In this paper, we extend the above inequality to polar derivative of a polynomial. Further, we also prove an improved version of above inequality into polar derivative.

TURÁN-TYPE Lr-INEQUALITIES FOR POLAR DERIVATIVE OF A POLYNOMIAL

  • Robinson Soraisam;Mayanglambam Singhajit Singh;Barchand Chanam
    • Nonlinear Functional Analysis and Applications
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    • 제28권3호
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    • pp.731-751
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    • 2023
  • If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into Lr-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

Synthesis and Electrorheological Effect of the Suspensions Composed of Nano Sized Hollow Polyaniline Derivatives

  • Choi Ung-Su
    • KSTLE International Journal
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    • 제7권1호
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    • pp.18-21
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    • 2006
  • The electrorheology of hollow PANI derivative suspensions in silicone oil was investigated by varying the electric fields and shear rates, respectively. The hollow PANI derivative susepnsions showed a typical electrorheological (ER) response caused by the polarizability of an amide polar group and shear yield stress due to the formation of chains upon application of an electric field. The shear stress for the hollow PANI succinate suspension exhibited an electric field power of 0.67. On the basis of the experimental results, the newly synthesized hollow PANI derivative suspensions were found to be an anhydrous ER fluid.

INEQUALITIES CONCERNING POLYNOMIAL AND ITS DERIVATIVE

  • Zargar, B.A.;Gulzar, M.H.;Akhter, Tawheeda
    • Korean Journal of Mathematics
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    • 제29권3호
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    • pp.631-638
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    • 2021
  • In this paper, some sharp inequalities for ordinary derivative P'(z) and polar derivative DαP(z) = nP(z) + (α - z)P'(z) are obtained by including some of the coefficients and modulus of each individual zero of a polynomial P(z) of degree n not vanishing in the region |z| > k, k ≥ 1. Our results also improve the bounds of Turán's and Aziz's inequalities.

Lr INEQUALITIES FOR POLYNOMIALS

  • Reingachan N;Mayanglambam Singhajit Singh;Nirmal Kumar Singha;Khangembam Babina Devi;Barchand Chanam
    • Nonlinear Functional Analysis and Applications
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    • 제29권2호
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    • pp.451-460
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    • 2024
  • If a0 + Σnν=μ aνzν, 1 ≤ µ ≤ n, is a polynomial of degree n having no zeroin |z| < k, k ≥ 1 and p'(z) its derivative, then Qazi [19] proved $$\max_{{\left|z\right|=1}}\left|p\prime(z)\right|\leq{n}\frac{1+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|k^{{\mu}+1}}{1+k^{{\mu}+1}+\frac{{\mu}}{n}\left|\frac{a_{\mu}}{a_0} \right|(k^{{\mu}+1}+k^{2{\mu}})}\max_{{\left|z\right|=1}}\left|p(z)\right|$$ In this paper, we not only obtain the Lr version of the polar derivative of the above inequality for r > 0, but also obtain an improved Lr extension in polar derivative.

INTEGRAL MEAN ESTIMATES FOR SOME OPERATOR PRESERVING INEQUALITIES

  • Shabir Ahmad Malik
    • Korean Journal of Mathematics
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    • 제32권3호
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    • pp.497-506
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    • 2024
  • In this paper, some integral mean estimates for the polar derivative of a polynomial with complex coefficients are proved. We will see that these type of estimates are new in this direction and discuss their importance with respect to existing results comparatively. In addition, the obtained results provide valuable insights into the behavior of integrals involving operator preserving inequalities.

INEQUALITIES FOR B-OPERATOR

  • Akhter, Rubia;Gulzar, M.H.
    • Korean Journal of Mathematics
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    • 제30권3호
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    • pp.525-532
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    • 2022
  • Let 𝓟n denote the space of all complex polynomials $P(z)=\sum\limits_{j=0}^{n}{\alpha}_jz^j$ of degree n. Let P ∈ 𝓟n, for any complex number α, DαP(z) = nP(z) + (α - z)P'(z), denote the polar derivative of the polynomial P(z) with respect to α and Bn denote a family of operators that maps 𝓟n into itself. In this paper, we combine the operators B and Dα and establish certain operator preserving inequalities concerning polynomials, from which a variety of interesting results can be obtained as special cases.